Proposition 11

Assume that R is irreducible. Let $\alpha$ and $\beta$ be two roots such that $\| \alpha \| = \| \beta \|$. There exists $g \in W(R)$ such that $g(\alpha) = \beta$.

The transforms of $\alpha$ by $W(R)$ generate V (no. 2, Cor. of Prop. 5). Hence there exists $g \in W(R)$ such that $(g(\alpha)|\beta) \neq 0$. We assume from now on that $(\alpha|\beta) \neq 0$. By formula (9) of no. 1, $n(\alpha, \beta) = n(\beta, \alpha)$. Replacing $\beta$ if necessary by $s_\beta(\beta) = -\beta$, we can assume that $n(\alpha, \beta) > 0$. Then, by the list at the beginning of no. 3, either $\alpha = \beta$ (in which case the proposition is clear), or $n(\alpha, \beta) = n(\beta, \alpha) = 1$; in that case